A convenient measure of the influence of an isolated star is its
Einstein radius (Equation 4). The optical depth for a source
to lie within an Einstein ring is given by
= */cr, where * is the total
stellar surface density, and is independent of the masses of the
individual stars. When << 1,
the faint microimages associated with
individual stars are unimportant except on the rare occasions when a
star crosses within a few E of the line of sight. When this
happens the source will brighten and fade on a timescale given by

As increases, the
microlenses can no longer be considered in
isolation. At moderate , the
combined, long range action
(through shear) of the background stars can be thought of as creating
a quadrupole lens at the site of each star
(Chang & Refsdal
1984,
Nityananda &
Ostriker 1984,
Lee & Spergel
1990).
This can break the
circular symmetry and allow slender caustic surfaces to form behind
individual stars, leading to extra image pairs. At still higher
, when the Einstein rings
start to overlap, a complicated
caustic network will develop
(Schneider &
Weiss 1986,
Kayser et
al. 1989,
Wambsganss 1990,
Witt 1990).
The backgound shear from the
largescale mass distribution of the lens makes the network
anisotropic and the variability will be sensitive to the direction of
transverse motion relative to the shear
(Wambsganss 1990,
Wambsganss et
al. 1990a).
Frequent, large amplitude image variation is possible in this regime
(Paczynski 1986a,
Nemiroff 1986,
Schneider &
Weiss 1987,
Witt 1990).
If there is a smooth, supercritical background
density, then dramatic demagnifications are also expected when the
ray associated with the brightest microimage intercepts a star. The
net effect will be to conserve flux so that the mean magnification is
the same as if the mass in the stars had been smoothed out
(Peacock 1986).
Since fold caustics dominate at the highest magnification, the
asymptotic probability for large magnification by more than µ
scales as µ-2
(Section 3.3,
Vietri &
Ostriker 1983,
Nityananda &
Ostriker 1984,
Blandford &
Narayan 1986),
with a normalization that can be computed exactly
(Schneider 1987c).
Numerical simulations show
a significant excess in the crosssection (over and above the
analytical normalization of the µ-2 law) at moderate
magnifications
(Rauch et
al. 1991).

When a caustic associated with the macroimage is approached, the
number of stars contributing to the macroimage as well as the mean
magnification increase, rendering numerical simulation impractical
(Deguchi &
Watson 1987,
1988).
For ~ 1, the number of
microimages becomes so large that the fluctuation level in fact
diminishes. When >> 1, the
mean magnification decreases -2
again but the relative fluctuations, for fixed
source size, are found to increase
(Deguchi &
Watson 1988).
In this limit the angular profile of the macroimage consists of a Gaussian
core and a power law tail -4
(Katz et
al. 1986).

Variability associated with microlensing is strongly attenuated if the
angular size of the source becomes comparable to the Einstein radius
of the microlenses (e.g.
Wambsganss 1992,
Refsdal &
Stabell 1991).
This can be used to set limits on the sizes of emission regions of
distant sources (cf Section 6.1).